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JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY
VOLUME 21
The Effects of Rainfall Inhomogeneity on Climate Variability of Rainfall Estimated
from Passive Microwave Sensors
CHRISTIAN KUMMEROW
Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado
PHILIP POYNER
USAF, Vandenberg AFB, California
WESLEY BERG
AND JODY
THOMAS-STAHLE
Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado
(Manuscript received 30 July 2003, in final form 7 November 2003)
ABSTRACT
Passive microwave rainfall estimates that exploit the emission signal of raindrops in the atmosphere are
sensitive to the inhomogeneity of rainfall within the satellite field of view (FOV). In particular, the concave
nature of the brightness temperature (T b ) versus rainfall relations at frequencies capable of detecting the blackbody
emission of raindrops cause retrieval algorithms to systematically underestimate precipitation unless the rainfall
is homogeneous within a radiometer FOV, or the inhomogeneity is accounted for explicitly. This problem has
a long history in the passive microwave community and has been termed the beam-filling error. While not a
true error, correcting for it requires a priori knowledge about the actual distribution of the rainfall within the
satellite FOV, or at least a statistical representation of this inhomogeneity. This study first examines the magnitude
of this beam-filling correction when slant-path radiative transfer calculations are used to account for the oblique
incidence of current radiometers. Because of the horizontal averaging that occurs away from the nadir direction,
the beam-filling error is found to be only a fraction of what has been reported previously in the literature based
upon plane-parallel calculations. For a FOV representative of the 19-GHz radiometer channel (18 km 3 28 km)
aboard the Tropical Rainfall Measuring Mission (TRMM), the mean beam-filling correction computed in this
study for tropical atmospheres is 1.26 instead of 1.52 computed from plane-parallel techniques. The slant-path
solution is also less sensitive to finescale rainfall inhomogeneity and is, thus, able to make use of 4-km radar
data from the TRMM Precipitation Radar (PR) in order to map regional and seasonal distributions of observed
rainfall inhomogeneity in the Tropics. The data are examined to assess the expected errors introduced into climate
rainfall records by unresolved changes in rainfall inhomogeneity. Results show that global mean monthly errors
introduced by not explicitly accounting for rainfall inhomogeneity do not exceed 0.5% if the beam-filling error
is allowed to be a function of rainfall rate and freezing level and does not exceed 2% if a universal beam-filling
correction is applied that depends only upon the freezing level. Monthly regional errors can be significantly
larger. Over the Indian Ocean, errors as large as 8% were found if the beam-filling correction is allowed to vary
with rainfall rate and freezing level while errors of 15% were found if a universal correction is used.
1. Introduction
Climate studies of rainfall trends and variability require satellite-based products in order to overcome the
extremely limited in situ observations over the world’s
oceans. Because passive microwave observations are
directly correlated to the amount of liquid water in the
rain column, methods such as those developed by Wilheit et al. (1991), Kummerow et al. (2001), or Petty
Corresponding author address: Christian Kummerow, Dept. of Atmospheric Science, Colorado State University, Fort Collins, CO
80523-1371.
E-mail: [email protected]
q 2004 American Meteorological Society
(1994) have been favored over infrared techniques that
have good temporal sampling but poor physical relations
with the actual rain. To make optimal use of both types
of sensors, the Global Precipitation Climatology Project
(GPCP) (Huffman et al. 1997) merges the passive microwave results with infrared data but only after using
the microwave results to remove any regional biases
from the infrared data. The utility of passive microwave
sensors can also be inferred from the ever-increasing
number of sensors. As of this writing, there are three
Special Sensor Microwave Imager (SSM/I) instruments
in orbit in addition to the Tropical Rainfall Measuring
Mission (TRMM) Microwave Imager (TMI), the Japanese Advanced Microwave Sounding Radiometer
APRIL 2004
KUMMEROW ET AL.
(AMSR-E) aboard the Aqua satellite, and a similar instrument aboard the Japanese Advanced Earth Observing Satellite-II (ADEOS-II). Despite their increasing
use, however, passive microwave estimates are not free
of uncertainties as they contain a number of implicit
assumptions related to cloud morphology and microphysical properties. To the extent that real cloud systems
deviate from these assumptions, one must expect errors
in the microwave products. These errors vary from random, which reflect deviations of individual clouds from
the norm, to ocean-basin- and global-scale variations
associated with large-scale changes in precipitation
cloud properties. It is hypothesized that these large-scale
changes in cloud properties are in response to largescale changes in the external forcing mechanisms as
might be evidenced during an El Niño–Southern Oscillation (ENSO) event.
Recent data from the TRMM satellite (Kummerow et
al. 2000) make it possible to examine large-scale differences in rainfall derived from two very distinct sensors—the TMI and the TRMM Precipitation Radar (PR).
While neither is considered ‘‘truth’’ in this study, their
differences can be studied at the large space and time
scales important for climate applications. Because the
TMI and PR sense very different aspects of precipitating
clouds, any systematic differences between these two
sensors are likely caused by changes in the underlying
cloud morphology or microphysics that cannot be directly observed by one or both sensors. Figure 1a shows
the temporal deviations of the mean tropical rainfall
derived from the two passive microwave and radar algorithms used by the TRMM project. The two passive
microwave algorithms are those of Wilheit et al. (1991)
designed for monthly 58 3 58 accumulations and Kummerow et al. (2001), which is optimized for instantaneous retrievals. The algorithms use the same input data
but are quite distinct in their philosophy. The Wilheit
et al. (1991) scheme, known as the TRMM 3A11 product, exploits the observed warming of brightness temperature T b at 19 GHz due to the blackbody emission
of raindrops over a radiometrically cold ocean. The
scheme uses a simple conceptual cloud model to construct the T b versus rainfall relations. While prone to
large errors at the pixel level, its strength is in its simplicity and the large reduction of random errors that
occurs for satellite rainfall estimates over large space
and time domains. In contrast, the Kummerow et al.
(2001) approach, known as the TRMM 2A12 product,
tries to optimize the pixel-level retrievals by fully accounting for all channels on the TRMM radiometer. This
is accomplished by a Bayesian inversion methodology
that introduces a priori information from a set of precomputed cloud-resolving model profiles. The added
complexity is beneficial for pixel-level retrievals, but
the long-term stability of it in the algorithm is more
difficult to verify. As such, the two approaches are highly complementary. The TRMM PR product, known as
2A25, uses a Z–R-type approach but modifies the orig-
625
FIG. 1. Time series for two TRMM radiometers and the TRMM
radar rainfall products for (a) the global oceans between 368N and
368S, and (b) a 208 3 208 area in the central Pacific.
inally assumed relation when the total path attenuation
is sufficiently robust to constrain the solution. As such,
the algorithm is susceptible to changes in the rainfall
drop size distribution, particularly for light to moderate
rainfall rates where the total path attenuation cannot be
distinguished from the background noise. Robertson et
al. (2003) have examined this issue and found some
evidence that changes in the drop size distribution might
indeed be occurring that are not being captured by the
PR algorithm.
Figure 1a shows deviations for each of the three algorithms discussed above from their own 4-yr climatologies. These variability plots clearly show that the
two passive microwave algorithms agree quite well in
their global trends, but the PR shows a significantly
reduced variability. The greatest differences appear at
the height of the El Niño event of 1997–98 at the beginning of the time series. At these large scales, sampling differences between TMI and PR are negligible,
and the details of the inversion approach do not appear
to be as important as the basic data used. As such, this
behavior is consistent with the hypothesis that these
differences are not caused by specific algorithm errors
but rather by the large-scale systematic changes in rain-
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JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY
fall morphology or microphysics that are assumed constant by one or both of the sensors. A difference of
approximately 0.3 mm day 21 , or 10% of the mean global
rainfall, however, has important consequences. Soden
(2000) found that this was the magnitude of disagreement between climate models and observations associated with the 1982–83 ENSO event. Yet, despite the
large differences in rainfall variability at the tropical
ocean scale shown in Fig. 1a, these differences are not
uniform in space. If one focuses on a 208 3 208 region
in the central Pacific between 1708E and 1708W and
6108 latitude, one finds substantially improved trend
correlations between the sensors as is shown in Fig. 1b.
Within the current hypothesis, such behavior would be
expected if only the total rainfall changed while the
cloud morphology and microphysics remain relatively
constant during this period.
In order to resolve what appear to be regional biases
and trends related to changes in cloud properties, assumed to be constant, a systematic analysis of potential
errors must be undertaken. Ground-based observations
over oceans, because of their limited spatial and temporal coverage, can hardly be expected to resolve differences that are regionally and temporally varying with
correlation lengths far larger than a ground-based system can observe. Instead, an error-characterization strategy is needed to build an independent error model for
rainfall estimates that considers the earth system in its
entirety. In this paradigm, instead of relying on groundbased comparisons, the assumptions in the algorithms
are examined one at a time until a comprehensive error
model can be put forth based on first principles. The
role of ground validation in this paradigm shifts from
one of providing point comparisons with satellite products to one in which they are used to verify the hypotheses and procedures used in the global error characterization.
Some assumptions in the algorithms can be shown to
have relatively little impact upon the computed radiances. In this case, one can allow a generous uncertainty
in the assumed parameter without introducing much uncertainty in the retrieval. In other cases, the assumed
variables affect retrievals in a substantial manner and
error propagation models lead to excessive uncertainties
unless the assumed parameter is somehow constrained
to reflect its actual variability instead of its potential
variability. Rainfall inhomogeneity is such a parameter.
The assumed rainfall inhomogeneity within relatively
large (10–60 km) footprints of current microwave radiometers can lead to significant errors in the retrieved
rainfall products. These errors are related to the nonlinear relation between the brightness temperatures, T b ,
and rainfall. The bias, resulting from nonuniform beam
filling, was first described by Wilheit (1986). While
Spencer et al. (1983) had also observed this bias, that
study focused only on the partial beam filling by uniform
rainshowers within the radiometer field of view (FOV)
and did not offer a coherent explanation of the effect.
VOLUME 21
Since then, numerous authors including Chiu et al.
(1990), Graves (1993), Petty (1994), Ha and North
(1995), North and Polyak (1996), and Kummerow
(1998) have all looked at this problem from theoretical
as well as statistical considerations. As such, the problem is thought to be theoretically well understood, and
in principle, the error propagation model is simple. Unfortunately, the rainfall variability is not a simple function of rainfall rate and can be shown (section 3) to vary
both regionally and temporally. As such, it is not simply
enough to understand the theoretical basis for the beamfilling error, but quantitative global statistics of rainfall
inhomogeneity at scales below current FOV sizes are
needed in order to assess the actual impact of this uncertainty upon rainfall products at various space and
time scales. The magnitude of the error introduced by
this uncertain subresolution variability will eventually
depend both upon the theoretical foundation of this error, as well as the extent to which the climate system
allows changes in rainfall variability to occur at various
time and space scales.
This study makes use of the relatively high resolution
TRMM PR data to serve as a proxy for the actual rainfall
variability within the TRMM TMI footprint, which is
18 km 3 30 km for the 19-GHz channels (approximated
by a square area of 24 km 3 24 km for computational
simplicity in this study). Before undertaking an examination of the observed variability and its consequences
for regional and global biases, however, section 2 of
this paper provides a brief review of the theoretical basis
of the beam-filling correction, including the effects of
using one-dimensional radiative transfer models to treat
a problem that is inherently three-dimensional. Section
3 examines PR-observed rainfall inhomogeneity to look
for both regional and interannual variation. A discussion
and conclusions are presented in section 4.
2. The beam-filling error
A passive microwave sensor’s ability to measure rainfall over oceans depends on the blackbody emission of
liquid drops, which offer a strong contrast to radiometrically cold ocean surfaces at these wavelengths. Irrespective of the inversion details, all physically based
retrieval algorithms begin with radiative transfer computations to establish relationships between an assumed
cloud structure and the satellite-observed T b . Wilheit
(1986) described the simple conceptual rainfall cloud
used in this study. In that cloud, a Marshall–Palmer
(Marshall and Palmer 1948) distribution of raindrops is
assumed from the surface up to the freezing level (08C
isotherm). Density of water is adjusted to keep a constant rainfall rate throughout the column. A standard
lapse rate of 6.5 K km 21 was specified and the relative
humidity was assumed to be 80% at the surface, increased linearly to 100% at the freezing level, and remained at 100% above that. A nonprecipitating cloud
layer containing 0.5 g m 23 of cloud liquid water is as-
APRIL 2004
627
KUMMEROW ET AL.
FIG. 2. Upwelling T b computed from plane-parallel theory for a 24
km 3 24 km FOV assuming a uniform rain field (solid curve) as
well as the actual observed sub-FOV inhomogeneity (dotted line).
sumed in the 0.5 km just below the freezing level, while
variable amounts of ice can be specified above the freezing level. Because it is not specified in the Wilheit
(1986) paper, we assume that the ice extends 3 km above
the freezing level with amounts equal to 0.75, 0.5, and
0.25 times the rainwater content in each successive kilometer above the freezing level. Ice is modeled as graupel with a density of 0.4 g cm 23 with a Marshall–Palmer
drop size distribution. These assumptions thus couple
the surface temperature, the freezing level, and the precipitable and cloud water contents of the atmosphere.
The rainfall intensity couples the total rainwater content
to the ice aloft. Specifying a surface temperature and a
rainfall rate determines all cloud parameters in the model.
a. Plane-parallel approximation
A plane-parallel Eddington approximation (Kummerow 1993) is used for the radiative transfer computations. The radiative approximation introduces errors
of less than 1–2 K for the frequencies examined in this
study (Smith et al. 2002). The effect of using a planeparallel computation to examine inhomogeneous rainfall, which is inherently three-dimensional, is examined
separately. The atmospheric model described above
specifies all the input parameters to the radiative transfer
equations except for the surface emissivity which, in
turn, depends upon the near-surface wind speed. Here,
we use a wind speed of 6 m s 21 to represent average
oceanic conditions. Oxygen and water vapor absorption
are computed from Liebe et al. (1993) while cloud water
absorption is specified by Rayleigh theory. The absorption in this regime is independent of particle sizes. Scattering parameters for raindrop and graupel particles are
computed from Mie (1908) theory.
Figure 2 shows the relations between theoretically
derived T b and area mean rainfall assuming either homogeneous rain (solid curve) or inhomogeneous rain
(dashed curve) where the actual rainfall distribution is
taken from 4-km PR data. The solid curve in Fig. 2
shows the computed T b corresponding to the cloud model described above for 19.35-GHz, vertically polarized
radiation with a view angle of 538 over an ocean background. The computations were made assuming a sea
surface temperature of 302 K (freezing level of 4.5 km)
thought to be representative of the Tropics. The dashed
line in the figure corresponds to the average T b in a 24
km 3 24 km FOV computed from 4-km TRMM PR
rainfall data. The appropriate model corresponding to
the PR surface rainfall is used for each PR pixel and
plane-parallel radiative transfer computations are performed for each 4 km 3 4 km pixel before the T b are
averaged to the radiometer FOV. The rainfall plotted on
the horizontal axis in this case is the FOV-averaged
rainfall. The fact that the T b curve corresponding to
nonuniform rainfall is always below the uniform rainfall
curve is predicated by the concave nature of the uniform
rainfall relation. Inversion schemes that attempt to invert
the radiance signal based upon uniform rainfall assumptions will therefore infer lower rainfall rates than
the true rainfall as indicated by the dashed line. Unlike
the uniform rainfall curve, which is fully determined by
the conceptual cloud, the nonuniform rainfall curve
shows considerable variability that results from different
realizations within a given FOV leading to the same
mean rainfall. The fact that the tops of the error bars
are typically well below the homogeneous rainfall curve
for all but the lightest rain cases (low T b ) implies that
homogeneous rainfall rarely occurs at a scale of 24 km
3 24 km. Mean T b and variances were computed for 3
months of oceanic PR data from December 1999
through February 2000. Only the six pixels on each side
of nadir were used for this purpose. All freezing levels
were assumed to be 4.5 km for these calculations.
Because of the bias introduced when homogeneous
rainfall is assumed in the radiative transfer computations
for rain fields that are not homogeneous, a beam-filling
correction can be defined to account for this bias if the
variability is known. The beam-filling correction was
obtained in this study by computing the T b from the
inhomogeneous FOV and inverting it to obtain a rainfall
rate following the homogeneous rainfall curve (i.e., solid line). The ratio of the true rainfall rate to that inferred
from the homogeneous rainfall curve constitutes the
beam-filling correction. Following Chiu et al. (1990)
the data were binned by mean rainfall, ^R&, and rainfall
inhomogeneity parameter, C, defined for a given satellite FOV by C 5 sR/^R&, where sR is the standard
deviation of the individual high-resolution measurements:
[
1
sR 5
N21
O (R 2 ^R&) ]
36
2
n
n51
0.5
.
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JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY
Here, N 5 36 because 4-km rain data within a 24 km
3 24 km satellite FOV are used. Table 1 presents the
actual values for the beam-filling correction as a function of mean rainfall rate, ^R&, and inhomogeneity parameter, C. Within each ^R& and C category, the standard deviation of the mean beam-filling correction is
quite small relative to changes in the rainfall categories.
This is consistent with the results of Chiu et al. (1990),
who showed from theoretical considerations that the
above two parameters should fully specify the beamfilling correction if the rainfall follows a well-defined
statistical distribution with the satellite FOV.
The beam-filling corrections shown in Table 1 are a
strong function of the mean rainfall rate as well as the
inhomogeneity parameter. Unfortunately, the inhomogeneity parameter cannot be determined by the radiometer itself. In order to develop a mean correction for
this effect, Chiu et al. (1990) used mean statistics from
4-km shipborne radar data obtained during the Global
Atmospheric Research Program (GARP) Atlantic Tropical Experiment (GATE) to compute corrections applicable to various FOV sizes. For a 24 km 3 24 km FOV,
their results indicate a mean correction of approximately
1.54 or 1.68 depending upon which intensive observing
period (IOP) was selected. This compares well to a mean
value of 1.52 computed from the PR data described
above if all raining pixels are used. A value of 1.60 is
obtained if only pixels with rainfall in excess of 0.5 mm
h 21 are used in the calculations. The 0.5 mm h 21 criterion corresponds roughly to the limit of detectability
of passive microwave radiometers. The mean corrections in both cases are computed by weighting the values, as computed in Table 1, with the appropriate distribution of observed ^R& and C in the 24 km 3 24 km
FOV. These corrections, however, assume a plane-parallel radiative transfer solution. If interactions between
neighboring clouds are allowed, the correction factors
will be seen in the next section to decrease rather substantially from the above values.
b. Slant-path approximation
The nature of the Eddington approximation is such
that diffuse radiances are computed first. Once the diffuse radiance is known, the upwelling T b are computed
by tracing individual rays through the cloud allowing
for emission, absorption, scattering out of the beam, and
scattering back into the direction of view from previously computed diffuse radiation. As such, the Eddington approximation is amenable to a pseudo-three-dimensional version in which the diffuse radiance is computed as if each pixel were independent and horizontally
infinite, but the ray tracing is done through the actual
three-dimensional structure of the cloud. This approximation was used by Bauer et al. (1998) who found
excellent agreement with a full three-dimensional backward Monte Carlo code developed previously by Roberti et al. (1994). The pseudo-three-dimensional Ed-
VOLUME 21
dington approximation, referred to as the slant-path approximation, is used here because of the prohibitive cost
of running the Monte Carlo code for 3 months of PR
data representing approximately 40 000 individual radiometer FOVs. For typical view angles around 538 used
by spaceborne radiometers, the slant path through the
raining clouds is expected to average radiation from
neighboring pixels and thus smooth out the upwelling
radiance field—particularly if horizontal dimensions are
significantly smaller than the height of the rain column.
Following the procedure of the previous section, Fig.
3 shows the T b computed from the slant-path approximation and compares it to the homogeneous rainfall
case. Because periodic boundary conditions are assumed by the slant-path code, the homogeneous rainfall
curve matches the plane-parallel result. The T b computed from the slant-path approximation, however, are
significantly closer to the homogeneous curve due to
the horizontal averaging that takes place as radiation
crosses the cloud along a slanted path. Alternatively,
the slanted-path averaging has the same effect as reducing the inhomogeneity parameter C. This can also
be confirmed in Table 2, which presents the quantitative
beam-filling corrections as a function of ^R& and C. The
corrections are smaller for each ^R& and C, as well as
for the mean global beam-filling correction. For the
slant-path approximation, the mean beam-filling correction is 1.26 when all raining pixels are considered,
and 1.29 if a rain threshold of 0.5 mm h 21 is assumed.
Qualitatively, this result is in agreement with a previous
study by Petty (1994), who used three-dimensional computations to conclude that radiation escaping from the
sides of clouds would lead to beam-filling corrections
much smaller than what is inferred from plane-parallel
calculations. Quantitatively, these values are consistent
with the TRMM operational algorithm developed by
Wilheit et al. (1991). The beam-filling correction of this
algorithm is based on slant-path calculations performed
by Wang (1996) using aircraft radar data obtained during
the Tropical Ocean Global Atmosphere Coupled Ocean–
Atmosphere Response Experiment (TOGA COARE).
For TMI, the correction factor is given by 1 1 0.062
3 freezing height (Chang and Chiu 2001), which results
in a value of 1.28 for a freezing height of 4.5 km used
in this study. Results are also consistent with aircraft
radar data from the Kwajalein Experiment (KWAJEX)
evaluated by Chen (2001) who found only a 4% difference between TOGA COARE and KWAJEX observations.
Because of differences between the plane-parallel and
slant-path approaches, there is the potential to significantly alter some of the current satellite rainfall products. There is, however, no clear path between the numbers presented here and the impact upon a specific algorithm. This is because most algorithms convolve the
beam-filling correction with other corrections and uncertainties. As such, the numbers presented here must
be analyzed in terms of each specific algorithm to assess
0–0.5
0.5–1
1–2
2–3
3–4
4–5
5–6
6–7
7–8
8–9
9–10
10–12
12–14
14–16
16–18
18–20
20–25
25–30
30–35
35–40
40–45
45–50
Rain rate
(mm h21 )
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
0.00
0.00
0.02
0.03
0.05
0.07
0.09
0.11
0.13
0.16
0.14
0.20
0.26
0.31
0.40
0.38)
0.39)
0.76
0.34)
0.13)
0.21)
0.5–1.0
1.00
1.00
1.02
1.07
1.13
1.19
1.25
1.30
1.37
1.43
1.46
1.57
1.72
2.00
2.36
(2.61
(2.58
3.63
(4.16
(4.72
(5.93
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
—)
0.00
0.01
0.01
0.01
0.02
0.02
0.03
0.04
0.05
0.05
0.08
0.08
0.04)
—)
—)
0.16)
0.0–0.5
(1.00
1.00
1.00
1.02
1.04
1.06
1.08
1.09
1.11
1.13
1.16
1.20
1.26
(1.42
(1.54
(1.66
(1.90
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
0.00
0.01
0.05
0.07
0.10
0.13
0.17
0.22
0.26
0.29
0.31
0.38
0.37
0.50
0.44
0.60
0.80
0.90
—)
0.32)
—)
1.00)
1.0–1.5
1.00
1
1.07
1.21
1.35
1.51
1.63
1.76
1.89
1.93
2.11
2.28
2.58
2.80
2.94
3.30
3.73
5.26
(5.01
(5.47
(6.54
(8.25
0.00
0.03
0.08
0.11
0.15
0.28
0.27
0.33
0.37
0.45
0.46
0.55
0.82
0.82
0.82
0.13)
1.37)
(7.34 6 —)
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
1.5–2.0
1.00
1.02
1.17
1.42
1.64
1.92
2.06
2.27
2.44
2.54
2.79
3.02
3.54
3.45
4.04
(3.37
(5.30
6
6
6
6
6
6
6
6
6
6
6
6
6
0.01
0.05
0.12
0.18
0.24
0.35
0.49
0.63
0.51
0.94
0.36)
1.12
0.87)
(4.97 6 0.58)
(7.64 6 2.21)
(7.20 6 1.64)
1.00
1.07
1.30
1.71
2.04
2.36
2.71
2.99
2.96
3.52
(3.06
3.75
(3.58
2.0–2.5
6
6
6
6
6
6
6
6
6
6
0.01
0.08
0.17
0.21
0.28
0.30
0.87)
1.08)
1.99)
0.09)
(4.97 6 —)
(3.26 6 —)
1.00
1.14
1.49
2.03
2.34
2.56
(3.56
(3.95
(4.97
(3.11
2.5–3.0
Inhomogeneity parameter
1.01
1.24
1.72
2.34
2.74
(3.09
(3.85
(4.43
(4.78
(5.27
6
6
6
6
6
6
6
6
6
6
0.02
0.11
0.22
0.26
0.51
0.68)
0.50)
0.08)
0.16)
—)
3.0–3.5
6
6
6
6
6
6
0.04
0.15
0.31
0.54
0.71)
0.02)
(2.91 6 —)
1.01
1.38
2.07
2.93
(3.36
(2.75
3.5–4.0
1.02
1.64
(2.57
(3.69
(4.62
(5.54
4.81
6
6
6
6
6
6
6
0.06
0.24
0.42)
0.64)
—)
—)
1.13
4.0–5.0
1.05 6 0.12
2.20 6 0.25
(3.28 6 0.12)
5.0–6.0
TABLE 1. Mean beam-filling correction factor and standard deviation computed as a function of the average rainfall, ^R&, and the inhomogeneity parameter, C, using plane-parallel radiative
transfer calculations. Entries in parentheses indicate fewer than 10 data points. An absent standard deviation (6 —) indicates that only one data point was available while blank entries
indicate that no data were available for that category.
APRIL 2004
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JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY
FIG. 3. Upwelling T b computed from slant-path radiative transfer
methods for a 24 km 3 24 km FOV assuming a uniform rain field
(solid curve) as well as the actual observed sub-FOV inhomogeneity
(dotted line).
the degree to which the algorithm would change with
the new beam-filling correction employed and the extent
to which other global assumptions can offset any changes in the mean beam-filling correction. The often arbitrary shape of the vertical distribution of rainwater
with height, for instance, could easily lead to higher or
lower surface rainfall rates for the same retrieved liquid
water content.
c. Sensitivity to data resolution
TRMM radar data at 4 km are available over the entire
Tropics. As such, they are ideally suited to study the
global behavior of the beam-filling correction. The data
cannot, however, address the question of whether the
data itself have sufficient spatial resolution to properly
quantify the beam-filling correction. To address this,
250-m-resolution ground-based radar data were collected as part of the AMSR-E rainfall validation effort
in March–April 2001 at Wallops Island, Virginia (Matrosov et al. 2002). While this dataset is limited in its
spatial and temporal domain, it can be used to study the
behavior of the beam-filling correction when higherresolution data are available. The rainfall products were
produced from the dual-polarization X-band radar belonging to the National Oceanic and Atmospheric Administration’s Environmental Technology Laboratory
(NOAA/ETL) using standard polarimetric techniques
out to a range of 40 km from the radar (Matrosov et al.
2002).
Based on the physical principles discussed by Chiu
et al. (1990) and the results from previous sections, one
would expect the mean rainfall, ^R&, and variability parameter, C, to fully determine the beam-filling correction regardless of the initial resolution of the data. This
is confirmed in Table 3, which shows a subset of the
mean beam-filling correction computed from the Wal-
VOLUME 21
lops radar data for rainfall rates in the range of 3–4 mm
h 21 and inhomogeneity parameters in the range of 0–
3.0. The table was generated for various initial radar
data resolutions in which values lower than 250 m were
obtained by simply averaging the initial 250-m data.
For comparison purposes, the freezing level was assumed to be the same 4.5 km value assumed earlier.
Values are within the standard deviation of those found
using PR data in the previous sections. However, the
inhomogeneity parameter itself is somewhat different
depending upon the original resolution of the data. This
can be seen in Table 3 by examining the number of
occurrences (in parentheses) corresponding to each rainfall inhomogeneity category. As the original resolution
decreases, more and more pixels fall into lower inhomogeneity categories as would be expected since the
inhomogeneity must be zero in the limit that 24-km data
are used as the initial measurement.
Table 4 shows the total impact of changing the original resolution upon the beam-filling correction when
averaged over the observed inhomogeneity parameter.
Because the beam-filling correction does not change significantly as a function of ^R& and C, the table primarily
reflects changes in the computed inhomogeneity parameter. In the plane-parallel case, the gradual increase in
the computed inhomogeneity with resolution is seen to
have a substantial effect upon the mean beam-filling
correction for the rainfall rates (2–6 mm h 21 ) presented.
The beam-filling correction captured by 4-km data can
be seen to be only 57% of the true beam-filling correction for the rainfall category 2–3 mm h 21 , 60% for 3–
4 and 4–5 mm h 21 , and 70% for the 5–6 mm h 21 category. In contrast to the plane-parallel calculations, this
effect has all but disappeared for the slant-path calculations and the beam-filling correction computed from
4-km data is seen to faithfully reproduce the 250-m
inferred corrections. Only minimal changes can be observed in going from 250-m data to 4-km data in the
slant-path calculations. The fact that it appears to be
completely insensitive to the original resolution may be
due to the lack of robust statistics from a single field
experiment once rainfall is stratified by mean rainfall
and inhomogeneity parameter. Even with more robust
statistics, however, the change in the computed beamfilling correction with increased resolution should be
significantly smaller than it is for the plane-parallel approximation.
d. Sensitivity to freezing level
The beam-filling correction, because it has its origin
in the blackbody emission of liquid water drops, is not
sensitive to the surface rainfall, but to the integrated
liquid water. As such, one can speak of the sensitivity
to rainfall only because the conceptual cloud model defined by Wilheit (1986) couples the surface rainfall to
the integrated water content. For lower freezing levels,
however, the previously computed results, assuming a
0–0.5
0.5–1
1–2
2–3
3–4
4–5
5–6
6–7
7–8
8–9
9–10
10–12
12–14
14–16
16–18
18–20
20–25
25–30
30–35
35–40
40–45
45–50
Rain rate
(mm h21 )
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
0.00
0.00
0.02
0.03
0.05
0.07
0.09
0.11
0.13
0.16
0.14
0.20
0.26
0.31
0.40
0.38)
0.39)
0.76
0.34)
0.13)
0.21)
0.5–1.0
1.00
1.00
1.02
1.07
1.13
1.19
1.25
1.30
1.37
1.43
1.46
1.57
1.72
2.00
2.36
(2.61
(2.58
3.63
(4.16
(4.72
(5.93
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
—)
0.00
0.01
0.01
0.01
0.02
0.02
0.03
0.04
0.05
0.05
0.08
0.08
0.04)
—)
—)
0.16)
0.0–0.5
(1.00
1.00
1.00
1.02
1.04
1.06
1.08
1.09
1.11
1.13
1.16
1.20
1.26
(1.42
(1.54
(1.66
(1.90
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
0.00
0.01
0.05
0.07
0.10
0.13
0.17
0.22
0.26
0.29
0.31
0.38
0.37
0.50
0.44
0.60
0.80
0.90
—)
0.32)
—)
1.00)
1.0–1.5
1.00
1
1.07
1.21
1.35
1.51
1.63
1.76
1.89
1.93
2.11
2.28
2.58
2.80
2.94
3.30
3.73
5.26
(5.01
(5.47
(6.54
(8.25
0.00
0.03
0.08
0.11
0.15
0.28
0.27
0.33
0.37
0.45
0.46
0.55
0.82
0.82
0.82
0.13)
1.37)
(7.34 6 —)
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
1.5–2.0
1.00
1.02
1.17
1.42
1.64
1.92
2.06
2.27
2.44
2.54
2.79
3.02
3.54
3.45
4.04
(3.37
(5.30
6
6
6
6
6
6
6
6
6
6
6
6
6
0.00
0.05
0.12
0.18
0.24
0.35
0.49
0.63
0.51
0.94
0.36)
1.12
0.87)
(4.97 6 0.58)
(7.64 6 2.21)
(7.20 6 1.64)
1.00
1.07
1.30
1.71
2.04
2.36
2.71
2.99
2.96
3.52
(3.06
3.75
(3.58
2.0–2.5
6
6
6
6
6
6
6
6
6
6
0.01
0.08
0.17
0.21
0.28
0.30
0.87)
1.08)
1.99)
0.09)
(4.97 6 —)
(3.26 6 —)
1.00
1.14
1.49
2.03
2.34
2.56
(3.56
(3.95
(4.97
(3.11
2.5–3.0
Inhomogeneity parameter
1.01
1.24
1.72
2.34
2.74
(3.09
(3.85
(4.43
(4.78
(5.27
6
6
6
6
6
6
6
6
6
6
0.02
0.11
0.22
0.26
0.51
0.68)
0.50)
0.08)
0.16)
—)
3.0–3.5
6
6
6
6
6
6
0.04
0.15
0.31
0.54
0.71)
0.02)
(2.91 6 —)
1.01
1.38
2.07
2.93
(3.36
(2.75
3.5–4.0
1.02
1.64
2.57
(3.69
(4.62
(5.54
4.81
6
6
6
6
6
6
6
0.06
0.24
0.42
0.64)
—)
—)
1.13
4.0–5.0
1.05 6 0.12
2.20 6 0.25
(3.28 6 0.12)
5.0–6.0
TABLE 2. Mean beam-filling correction factor and standard deviation computed as a function of the average rainfall, ^R&, and the inhomogeneity parameter, C, using slant-path radiative
transfer calculations. Entries in parentheses indicate fewer than 10 data points. An absent standard deviation (6 —) indicates that only one data point was available while blank entries
indicate that no data were available for that category.
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TABLE 3. Mean beam-filling corrections as a function of the initial
data resolution and rainfall inhomogeneity for rain rates between 3
and 4 mm h21 . The number of observations in each category is given
in parentheses.
Rainfall inhomogeneity parameter
Data
resolution 0.0–0.5 0.5–1.0 1.0–1.5 1.5–2.0 2.0–2.5 2.5–3.0
250
500
1
2
4
8
m
—
m
—
km
—
km
—
km
—
km 1.04 (1)
1.14
1.13
1.18
1.17
1.16
1.12
Plane parallel
(2) 1.33 (8) 1.57 (1)
—
2.36 (1)
(2) 1.31 (8) 1.55 (1)
—
2.31 (1)
(4) 1.30 (6) 1.51 (1) 2.21 (1)
—
(4) 1.28 (7)
—
2.01 (1)
—
(7) 1.25 (4) 1.73 (1)
—
—
(9) 1.30 (2)
—
—
—
250
500
1
2
4
8
m
—
m
—
km
—
km
—
km
—
km 1.03 (1)
1.07
1.07
1.06
1.06
1.05
1.03
(2)
(2)
(4)
(4)
(7)
(9)
Slant path
1.04 (8) 1.15 (1)
—
1.26 (1)
1.04 (8) 1.15 (1)
—
1.25 (1)
1.04 (6) 1.15 (1) 1.24 (1)
—
1.06 (7)
—
1.23 (1)
—
1.09 (4) 1.20 (1)
—
—
1.09 (2)
—
—
—
4.5-km freezing level, are no longer valid as these were
coupled to the total liquid water content of a 4.5-km
liquid column. Figure 4 shows the beam-filling correction for an inhomogeneity parameter of 1.0–1.5 with
(panel a) the plane-parallel approximation and (panel b)
the slant-path approximation as a function of the freezing level. As can be seen, the beam-filling correction
decreases with decreasing freezing height for both approximations.
Because the conceptual cloud defined by Wilheit
(1986) and used here has a constant rain rate in the
column, the liquid water content scales nearly linearly
with freezing height. The beam-filling correction for a
10 mm h 21 FOV with a 5-km freezing level should thus
be very similar to a 20 mm h 21 FOV with a 2.5-km
freezing level. Examination of Fig. 4 shows that both
TABLE 4. Mean beam-filling correction as a function of the initial
data resolution and rainfall rate for plane-parallel and slant-path radiative transfer schemes. Results have been averaged over the observed inhomogeneity for each table entry.
Data
resolution
Rainfall rate (mm h21 )
2–3
3–4
250
500
1
2
4
8
m
m
km
km
km
km
1.14
1.13
1.12
1.10
1.08
1.04
Plane parallel
1.40
1.38
1.36
1.30
1.24
1.14
250
500
1
2
4
8
m
m
km
km
km
km
1.02
1.02
1.02
1.02
1.02
1.01
Slant path
1.07
1.07
1.07
1.07
1.07
1.04
4–5
5–6
1.46
1.45
1.42
1.37
1.28
1.18
1.46
1.44
1.43
1.39
1.32
1.21
1.11
1.11
1.11
1.11
1.10
1.06
1.12
1.12
1.12
1.12
1.11
1.05
FIG. 4. Mean beam-filling corrections computed as a function of
rainfall rate and freezing level. All curves are for an inhomogeneity
parameter between 1.0 and 1.5: (a) plane-parallel and (b) slant-path
computations.
conditions yield a beam-filling correction of almost exactly 2.4. As such, the correction for lower freezing
height can be simply inferred from any one height and
the shape of the rain profile. Unfortunately, this is only
approximately true for the slant-path approximation. As
the freezing level decreases, so does the horizontal averaging that occurs when the slanted rays no longer cross
neighboring pixels. This is easy to visualize for a 4-km
PR pixel with a 500-m freezing level. Even at 508 incidence, only a small portion of the 4-km radiance will
actually cross into neighboring pixels. As such, the
slant-path and plane-parallel approximations converge
in the limit that the freezing level is much smaller than
the horizontal resolution of the data. This can be verified
by comparing Figs. 4a and 4b for the lowest freezing
level of 2.0 km.
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633
Unfortunately, these results also imply that, for lower
freezing levels, the slant-path calculations may also
have some sensitivity to the initial resolution of the data
as was shown in the previous section for plane-parallel
calculations. While this issue cannot be addressed with
the current data, the dependence should be relatively
small since low freezing levels have generally small
beam-filling corrections.
e. Sensitivity to cloud profiles
The previous section discussed the sensitivity of the
beam-filling correction upon the liquid water column.
The sensitivity to the ice loading can also be investigated. For this, the 4.5-km freezing level used in sections 2b–d is used again, but the ice amount is scaled
to produce clouds with 25%, 50%, 200%, and 400% of
the original clouds. Figure 5 shows the effect on the
beam-filling correction for the same variability interval
(1.0–1.5) as in the previous section. As can be seen
from Fig. 5, the effect is quite small—especially considering that for a FOV of 24 km, 90% of the rain falls
at rainfall rates less than 12 mm h 21 .
3. Global beam-filling corrections
Because the results of section 2 indicate that the mean
beam-filling correction can be accurately predicted for
the slant-path approximation using 4-km TRMM PR
data, it is quite straightforward to use the results from
Table 2 (plus equivalent tables for lower freezing levels)
to compute beam-filling corrections that should be applied to 24-km TRMM radiometer FOVs on a pixel by
pixel basis. Since beam-filling errors are not the only
source of uncertainty, however, this study does not attempt any actual retrievals but instead uses the TRMM
PR rainfall as the ‘‘true’’ rainfall and compares this to
errors that would have resulted from the beam-filling
uncertainty if high-resolution data were not available.
The mean corrections applied when high-resolution data
are not included are those derived in the earlier section
from 3 months of oceanic TRMM PR data from December 1999 through February 2000. The freezing-level
height dependence was incorporated in a straightforward manner by using the observed sea surface temperature (SST) and the same nominal lapse rate of 6.5
K km 21 used in the cloud profile.
In a pure radiometer algorithm, which was the only
option before the launch of the TRMM satellite in 1997,
or for radiometers other than the TMI, the subpixel inhomogeneity cannot be determined directly. In this case,
it is necessary to apply a mean beam-filling correction
based upon the inferred rainfall rate and freezing level
only. This is defined as the ‘‘RR 1 FL’’ solution and is
computed by using the average inhomogeneity parameter as a function of rain rate. This value is simply the
average over the different inhomogeneities for each rain
rate listed in Table 2 (and similar tables for different
FIG. 5. Mean beam-filling corrections computed as a function of
rainfall rate and a constant multiple of the original ice water content.
All curves are for an inhomogeneity parameter between 1.0 and 1.5:
(a) plane-parallel and (b) slant-path computations.
freezing levels) weighted by the probability of occurrence of each inhomogeneity in the training dataset. It
represents the best value that a radiometer could retrieve
on its own, without additional information regarding the
actual variability within any given FOV. In addition to
the RR 1 FL solution, there is an additional simplification that is possible if only the freezing level is used
to compute a mean correction. This solution is the one
implemented in the oceanic component of the microwave algorithm used in GPCP (Wilheit et al. 1991) as
well as in TRMM 3A11 and AMSR-E. This simplification was necessary because the algorithm uses a histogram-based approach that does not explicitly retrieve
a rainfall rate at each pixel. This solution is referred to
as the ‘‘FL only’’ solution and is computed as the rainfall-weighted average over all rainfall rates and vari-
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VOLUME 21
FIG. 6. Mean rainfall rates for MAM98 from (a) TRMM PR (assumed to be truth in this study), (b) a
perfect radiometer algorithm that uses rainfall and freezing level to correct for sub-FOV rainfall inhomogeneity effects, and (c) a perfect radiometer algorithm that uses only the freezing level to correct for subFOV rainfall inhomogeneity effects. (d) Differences between (a) and (b); (e) differences between (a) and (c).
abilities listed in Table 2 (and equivalent tables for different freezing levels).
The above procedure computes the mean beam-filling
correction for each pixel. The rainfall for each approximation is derived by dividing the true rainfall from the
TRMM PR by the actual correction determined from
the freezing level, rain rate, and rainfall variability, and
then multiplying this value by either the RR 1 FL or
the FL-only correction. Rain rates can then be accu-
mulated over any space and time scale to examine potential errors at those scales. Here, the emphasis is on
climate rainfall variability and seasonal averages over
3-month periods. Random effects are minimal at these
large scales. Figures 6a–c show the true rainfall and the
RR 1 FL solution, as well as the FL-only solution for
the period of March–May of 1998 (MAM98). To first
order, the three rainfall accumulation maps appear quite
similar. This is not surprising since the pixel-level cor-
APRIL 2004
KUMMEROW ET AL.
rections, while quite variable, have been averaged over
a significant number of realizations to create monthly
maps. The total rainfall accumulations of the three maps
shown in Figs. 6a–c are remarkably constant. The total
rain in the RR 1 FL solution is 1.5% lower than the
true rainfall while the total rain in the FL-only solution
is 0.9% higher. This is quite remarkable given that the
beam-filling correction values were computed from only
the center pixel positions of the TRMM PR during the
December 1999–February 2000 period. Such a constant
result appears to indicate that both the rainfall variability, which is responsible for variation in the RR 1
FL map, as well as the conditional rainfall rate, which
is responsible for most of the differences in the FL-only
map, remain relatively constant on a global scale.
Despite the small differences at the global scale, larger regional differences are noticeable. Figures 6d and
6e display the differences between the true rain and the
RR 1 FL solution as well as the FL-only solution, respectively. As can be seen, significant differences exist
over large regions. Red areas in Fig. 6d indicate areas
where the RR 1 FL solution is lower than the true rain.
Given the general trend of increasing beam-filling corrections with increasing variability, red areas in Fig. 6d
are thus areas where the actual rainfall variability for
observed rainfall rates is greater than the mean value
[derived during December–February (DJF) 1999/2000].
Generally speaking, most of these areas appear concentrated in the Indian Ocean, the Maritime Continent, and
the ITCZ areas just west of South America and Africa.
In contrast to the difference maps of the RR 1 FL
solution, the FL-only solution shows a very distinct pattern. This pattern reflects, to first order, the regional
differences in conditional mean rain rates. Results from
Table 4, as well as earlier work by Chiu et al. (1990),
clearly show the dependence of the beam-filling correction upon rainfall rate. Since large accumulations can
be the result of either continuous light rain or occasional
heavy rain, these biases are also present in the rainfall
accumulations that do not consider these differences.
In keeping with the original goals of this paper, Fig.
7a examines the tropical oceanic mean rainfall variability that is artificially introduced into rainfall climatologies by an incomplete knowledge of the subpixel
variability for the two solutions discussed previously.
As was the case with the MAM98 period discussed previously, there is very little variability on the global
monthly scale examined in this figure. The magnitude
of the error introduced by the unknown rainfall inhomogeneity within a given radiometer FOV for the RR
1 FL solution (solid curve) is seen to be exceedingly
small, rarely exceeding 0.5% on a monthly global scale.
This implies that on a global monthly basis, variations
in the rainfall inhomogeneity do not change sufficiently
to cause differences between radar and radiometer rainfall products such as those examined in Fig. 1. The
dashed line in Fig. 7a shows the corresponding results
for the FL-only solution. Its variability is slightly larger,
635
FIG. 7. Monthly errors introduced into the global mean oceanic
rainfall (408N–408S) by incomplete knowledge of the sub-FOV inhomogeneity. Solid lines are for the correction that uses rain rates
and freezing level while dashed lines are for the correction that uses
only the freezing level. (a) Slant-path radiative calculations and (b)
plane-parallel calculations.
amounting to slightly less than 2% during the El Niño
period in early 1998. This again is driven primarily by
a slightly higher mean conditional rainfall rate during
this period.
Figure 7b shows the same results but using the planeparallel approximation to compute the beam-filling correction as described in section 2. The results are nearly
identical to those shown in Fig. 7a, except that the planeparallel assumption leads to slight increases in the magnitude of the monthly variations. Still, even for the El
Niño period in early 1998, the global biases introduced
by the FL-only correction (dashed line) and plane-parallel assumption amount to no more than approximately
2%.
Regionally, however, biases introduced by the unknown rainfall inhomogeneity can be significantly larger. If one concentrates on the Indian Ocean, for instance,
one can see a significantly different picture. Figures 8a
and 8b show the same results as those in Fig. 7, but for
an area between 08 and 208N and 708 and 1108E intended
to represent the Indian Ocean. Unlike the global mean
values, this region is seen to vary far more significantly
and systematically over the 4-yr time period. The slant
RR 1 FL solution is seen to underestimate the true
rainfall by about 2.5% on average, but with monthly
variations of up to 10%. More dramatic is the bias introduced by the plane-parallel, FL-only solution. Here,
the mean bias over the Indian Ocean is approximately
10% but with a monthly deviation of another 10%–15%.
From a practical point of view, it is thus possible to
make significant regional errors, particularly if less desirable corrections such as the plane-parallel FL-only
solution are implemented.
In order to get a more complete picture of regions
that might be susceptible to climate-scale biases, Fig. 9
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JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY
FIG. 8. Monthly errors for the Indian Ocean (08–208N, 1308–1708E)
by incomplete knowledge of the sub-FOV inhomogeneity. Solid lines
are for the correction that uses rain rates and freezing level while
dashed lines are for the correction that uses only the freezing level.
(a) Slant-path radiative calculations and (b) plane-parallel calculations.
VOLUME 21
shows difference maps similar to those shown in Fig.
6 for MAM98, but averaged over the entire 44-month
time period from December 1997 through July 2001
examined in this study. This time period represents
TRMM data while the satellite was at an altitude of 350
km. Data after July 2001 are from a 400-km altitude
and were not used because they have slightly lower
spatial resolution that could impact the conclusions. Results from the 44 months are quite similar to those found
in the MAM98 period. For the RR 1 FL solution, there
is a low bias in the Indian Ocean, the Maritime Continent, the Caribbean, and the western extensions of the
ITCZ from South America and Africa. This may be a
result of the somewhat more continental nature of the
convection as it moves from the continents into the sea.
This is illustrated even more dramatically with the FLonly correction where all the areas of underestimation
(red) are in close proximity to the continents where
higher conditional rainfall rates are known to exist.
4. Summary and discussions
Rainfall validation has historically involved comparison of satellite estimates with a set of observations
made from the ground in order to gain confidence as
well as to assign error statistics to satellite products.
FIG. 9. (a) Mean rainfall rates for Dec 1998–Jun 2002 from TRMM PR (assumed to be truth in this study).
(b) The differences between the ‘‘true’’ rainfall and a perfect radiometer algorithm that uses rainfall and
freezing level to correct for sub-FOV rainfall inhomogeneity effects. (c) Same as in (b) but for a radiometer
algorithm that uses only the freezing level to correct for sub-FOV rainfall inhomogeneity effects.
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KUMMEROW ET AL.
This paradigm has problems. First, it has proven to be
extremely difficult to make accurate ground-based measurements of precipitation. Sparse gauges often correlate
poorly with satellite observations due to large discrepancies between the space and time scales of the two
instruments. More important, however, may be the fact
that algorithm errors are not constant in space and time,
but have a regional and time-dependent character. Validation sites, unless they account for this variability, will
observe only one point of what is in reality a complex
tapestry of biases that are introduced by observations
that are not fully constrained and are thus susceptible
to errors due to changing cloud characteristics. Without
complete knowledge of what these changing cloud characteristics may be and how they affect rainfall retrieval
schemes, it is therefore very difficult to have confidence
in the representativeness of a small number of groundbased comparisons.
The above issues have led to a physical validation
approach, which attempts to explicitly model the uncertainty in satellite rain products by examining each
assumption separately and then using error propagation
methods to derive the final uncertainty. Within this
framework, the aliasing of derived rainfall due to subFOV rainfall inhomogeneity has been identified as a key
parameter that was theoretically well understood but for
which the uncertainty at various time and space scales
had not been quantified. To that end, the current study
reviewed the theoretical basis for the correction and then
examined TRMM PR data as the source of sub-FOV
rainfall data to assess the impact of this rainfall inhomogeneity upon climate-scale rainfall errors.
Using PR data, the beam-filling correction was found
to closely match previous studies that used a planeparallel approach for computing T b based upon 4-km
ground-based radar data. The differences between an
earlier study by Chiu et al. (1990) showed greater variations between two periods of the GATE experiment
than with the current study using 3 months of global 4km TRMM PR data. When plane-parallel calculations
were replaced by more correct slant-path calculations,
however, the beam-filling bias was reduced from a factor
of approximately 1.52 to a value of 1.26. This effectively reduces the total uncertainty attributable to the
beam-filling uncertainty.
From an error modeling perspective, it was shown
that individual pixels could have very large variations
due to the unknown rainfall inhomogeneity. Based upon
Table 2, for instance, a Tb consistent with 5 mm h 21 in
the case of homogeneous rainfall is also consistent with
5 3 3.56 5 17.8 mm h 21 if the inhomogeneity parameter
is in the 2.5–3.0 range. A single pixel thus has an uncertainty of this magnitude but such calculations say
little about how these errors should be propagated to
larger time and space scales unless direct measurements
or proxy variables can be found to examine this question.
Because results from section 2 indicated that 4-km
data have sufficient resolution (when slant-path calculations are used) to fully describe the beam-filling correction, the TRMM PR was used to analyze variations
at long time and space scales. The 44-month average
bias was shown in Fig. 9. The figure clearly shows that
areas of systematically larger than average inhomogeneity exist in those regions that are thought to have some
continental traits. This is particularly apparent in the
correction that uses only the freezing level. This correction is susceptible to changes in both the mean conditional rain as well as to changes in the rainfall inhomogeneity within a given rain-rate category.
Perhaps a bit surprising, however, was the fact that
globally the beam-filling correction does not appear to
vary by more than a few percent irrespective of the
analysis method. Thus, while regional variations, as analyzed over a 208 3 408 area encompassing the Indian
Ocean, can be quite dramatic and aliased relative to the
global mean, globally these changes appear to cancel to
produce a very robust signal. The question raised in the
introduction regarding the possibility that beam-filling
biases might be responsible for the observed differences
between TRMM radar and radiometer products can now
be answered. Beam-filling biases cannot be responsible
for this discrepancy as Fig. 7 clearly shows that actual
changes in rainfall variability cannot account for more
than a 2.5% change in global oceanic rainfall, while the
ENSO discrepancy is on the order of 10%.
Despite the success of the current study in defining
uncertainties due to rainfall variability at different space
and time scales, the problem is of course, not solved.
The rainfall inhomogeneity is but one of the sources of
uncertainty that must be considered. Other sources of
uncertainty, such as the shape of the rainwater profile,
the amount of cloud water, the excess absorption caused
by melting particles, and the density and drop spectra
of ice particles, to name the most important sources of
uncertainty, all need to be analyzed in a similar fashion.
Some of these will not have readily accessible sources
of data. In these cases, a proxy variable will have to be
determined from ground observations first before a
global analysis and a corresponding global error model
can be constructed. Nonetheless, these regional biases
need to be understood. Without them, regional biases
determined from ground-based sensors (such as perhaps
a low bias in the Indian Ocean) cannot be put into the
proper global framework needed to fully assess global
satellite-derived rainfall products.
Acknowledgments. This work was supported by the
TRMM program under NASA TRMM Grants NAG511189 and NAG-13694.
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